Chapter

Advanced Metatheory

Stewart Shapiro

in Foundations without Foundationalism

Published in print March 2000 | ISBN: 9780198250296
Published online November 2003 | e-ISBN: 9780191598388 | DOI: http://dx.doi.org/10.1093/0198250290.003.0006
 Advanced Metatheory

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After setting out some issues concerning the meta‐language, this chapter presents a number of results about second‐order languages with standard semantics. Since the membership relation can be captured in a second‐order language, it is shown that, in a sense, nth‐order logic, when n >2, is reducible to second‐order logic. Next, plausible reflection principles are articulated, which concern the use of the set‐theoretic hierarchy as the background for model‐theoretic semantics. These imply the existence of so‐called ‘small’ large cardinals (e.g., inaccessibles, Mahlo cardinals). Analogues of the Löwenheim–Skolem theorems broach the realm of (large) large cardinals. The characterization of first‐order logic and various notions of definability are covered as well.

Keywords: cardinals; definability; inaccessible; large cardinals; Mahlo; membership; meta‐theory; reflection; Löwenheim‐Skolem

Chapter.  19636 words. 

Subjects: Philosophy of Mathematics and Logic

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